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Theorem ipasslem1 26160
Description: Lemma for ipassi 26170. Show the inner product associative law for nonnegative integers. (Contributed by NM, 27-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  U )
ip1i.9  |-  U  e.  CPreHil
OLD
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem1
Dummy variables  j 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0cn 10846 . . . . . . . . . . 11  |-  ( k  e.  NN0  ->  k  e.  CC )
2 ax-1cn 9580 . . . . . . . . . . . 12  |-  1  e.  CC
3 ip1i.9 . . . . . . . . . . . . . 14  |-  U  e.  CPreHil
OLD
43phnvi 26145 . . . . . . . . . . . . 13  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . . . . . . . . . 14  |-  X  =  ( BaseSet `  U )
6 ip1i.2 . . . . . . . . . . . . . 14  |-  G  =  ( +v `  U
)
7 ip1i.4 . . . . . . . . . . . . . 14  |-  S  =  ( .sOLD `  U )
85, 6, 7nvdir 25940 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  (
k  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
94, 8mpan 668 . . . . . . . . . . . 12  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  A  e.  X )  ->  (
( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
102, 9mp3an2 1314 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
111, 10sylan 469 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G ( 1 S A ) ) )
125, 7nvsid 25936 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
134, 12mpan 668 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  (
1 S A )  =  A )
1413adantl 464 . . . . . . . . . . 11  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1 S A )  =  A )
1514oveq2d 6294 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k S A ) G ( 1 S A ) )  =  ( ( k S A ) G A ) )
1611, 15eqtrd 2443 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 ) S A )  =  ( ( k S A ) G A ) )
1716oveq1d 6293 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) G A ) P B ) )
18 ipasslem1.b . . . . . . . . . . . . 13  |-  B  e.  X
19 ip1i.7 . . . . . . . . . . . . . 14  |-  P  =  ( .iOLD `  U )
205, 19dipcl 26039 . . . . . . . . . . . . 13  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
214, 18, 20mp3an13 1317 . . . . . . . . . . . 12  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
2221mulid2d 9644 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2322adantl 464 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( 1  x.  ( A P B ) )  =  ( A P B ) )
2423oveq2d 6294 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
255, 7nvscl 25935 . . . . . . . . . . . 12  |-  ( ( U  e.  NrmCVec  /\  k  e.  CC  /\  A  e.  X )  ->  (
k S A )  e.  X )
264, 25mp3an1 1313 . . . . . . . . . . 11  |-  ( ( k  e.  CC  /\  A  e.  X )  ->  ( k S A )  e.  X )
271, 26sylan 469 . . . . . . . . . 10  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( k S A )  e.  X )
285, 6, 7, 19, 3ipdiri 26159 . . . . . . . . . . 11  |-  ( ( ( k S A )  e.  X  /\  A  e.  X  /\  B  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
2918, 28mp3an3 1315 . . . . . . . . . 10  |-  ( ( ( k S A )  e.  X  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3027, 29sylancom 665 . . . . . . . . 9  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) G A ) P B )  =  ( ( ( k S A ) P B )  +  ( A P B ) ) )
3124, 30eqtr4d 2446 . . . . . . . 8  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( ( k S A ) G A ) P B ) )
3217, 31eqtr4d 2446 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) ) )
33 oveq1 6285 . . . . . . 7  |-  ( ( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  ->  (
( ( k S A ) P B )  +  ( 1  x.  ( A P B ) ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3432, 33sylan9eq 2463 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
35 adddir 9617 . . . . . . . . 9  |-  ( ( k  e.  CC  /\  1  e.  CC  /\  ( A P B )  e.  CC )  ->  (
( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
362, 35mp3an2 1314 . . . . . . . 8  |-  ( ( k  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
371, 21, 36syl2an 475 . . . . . . 7  |-  ( ( k  e.  NN0  /\  A  e.  X )  ->  ( ( k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3837adantr 463 . . . . . 6  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
k  +  1 )  x.  ( A P B ) )  =  ( ( k  x.  ( A P B ) )  +  ( 1  x.  ( A P B ) ) ) )
3934, 38eqtr4d 2446 . . . . 5  |-  ( ( ( k  e.  NN0  /\  A  e.  X )  /\  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
4039exp31 602 . . . 4  |-  ( k  e.  NN0  ->  ( A  e.  X  ->  (
( ( k S A ) P B )  =  ( k  x.  ( A P B ) )  -> 
( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
4140a2d 26 . . 3  |-  ( k  e.  NN0  ->  ( ( A  e.  X  -> 
( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) )  ->  ( A  e.  X  ->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
42 eqid 2402 . . . . . 6  |-  ( 0vec `  U )  =  (
0vec `  U )
435, 42, 19dip0l 26045 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
444, 18, 43mp2an 670 . . . 4  |-  ( (
0vec `  U ) P B )  =  0
455, 7, 42nv0 25946 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
0 S A )  =  ( 0vec `  U
) )
464, 45mpan 668 . . . . 5  |-  ( A  e.  X  ->  (
0 S A )  =  ( 0vec `  U
) )
4746oveq1d 6293 . . . 4  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( ( 0vec `  U ) P B ) )
4821mul02d 9812 . . . 4  |-  ( A  e.  X  ->  (
0  x.  ( A P B ) )  =  0 )
4944, 47, 483eqtr4a 2469 . . 3  |-  ( A  e.  X  ->  (
( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) )
50 oveq1 6285 . . . . . 6  |-  ( j  =  0  ->  (
j S A )  =  ( 0 S A ) )
5150oveq1d 6293 . . . . 5  |-  ( j  =  0  ->  (
( j S A ) P B )  =  ( ( 0 S A ) P B ) )
52 oveq1 6285 . . . . 5  |-  ( j  =  0  ->  (
j  x.  ( A P B ) )  =  ( 0  x.  ( A P B ) ) )
5351, 52eqeq12d 2424 . . . 4  |-  ( j  =  0  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) )
5453imbi2d 314 . . 3  |-  ( j  =  0  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( 0 S A ) P B )  =  ( 0  x.  ( A P B ) ) ) ) )
55 oveq1 6285 . . . . . 6  |-  ( j  =  k  ->  (
j S A )  =  ( k S A ) )
5655oveq1d 6293 . . . . 5  |-  ( j  =  k  ->  (
( j S A ) P B )  =  ( ( k S A ) P B ) )
57 oveq1 6285 . . . . 5  |-  ( j  =  k  ->  (
j  x.  ( A P B ) )  =  ( k  x.  ( A P B ) ) )
5856, 57eqeq12d 2424 . . . 4  |-  ( j  =  k  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
k S A ) P B )  =  ( k  x.  ( A P B ) ) ) )
5958imbi2d 314 . . 3  |-  ( j  =  k  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( k S A ) P B )  =  ( k  x.  ( A P B ) ) ) ) )
60 oveq1 6285 . . . . . 6  |-  ( j  =  ( k  +  1 )  ->  (
j S A )  =  ( ( k  +  1 ) S A ) )
6160oveq1d 6293 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
( j S A ) P B )  =  ( ( ( k  +  1 ) S A ) P B ) )
62 oveq1 6285 . . . . 5  |-  ( j  =  ( k  +  1 )  ->  (
j  x.  ( A P B ) )  =  ( ( k  +  1 )  x.  ( A P B ) ) )
6361, 62eqeq12d 2424 . . . 4  |-  ( j  =  ( k  +  1 )  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( (
( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) )
6463imbi2d 314 . . 3  |-  ( j  =  ( k  +  1 )  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( ( k  +  1 ) S A ) P B )  =  ( ( k  +  1 )  x.  ( A P B ) ) ) ) )
65 oveq1 6285 . . . . . 6  |-  ( j  =  N  ->  (
j S A )  =  ( N S A ) )
6665oveq1d 6293 . . . . 5  |-  ( j  =  N  ->  (
( j S A ) P B )  =  ( ( N S A ) P B ) )
67 oveq1 6285 . . . . 5  |-  ( j  =  N  ->  (
j  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
6866, 67eqeq12d 2424 . . . 4  |-  ( j  =  N  ->  (
( ( j S A ) P B )  =  ( j  x.  ( A P B ) )  <->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
6968imbi2d 314 . . 3  |-  ( j  =  N  ->  (
( A  e.  X  ->  ( ( j S A ) P B )  =  ( j  x.  ( A P B ) ) )  <-> 
( A  e.  X  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) ) )
7041, 49, 54, 59, 64, 69nn0indALT 10999 . 2  |-  ( N  e.  NN0  ->  ( A  e.  X  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) ) )
7170imp 427 1  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527   NN0cn0 10836   NrmCVeccnv 25891   +vcpv 25892   BaseSetcba 25893   .sOLDcns 25894   0veccn0v 25895   .iOLDcdip 26024   CPreHil OLDccphlo 26141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-sum 13658  df-grpo 25607  df-gid 25608  df-ginv 25609  df-ablo 25698  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-nmcv 25907  df-dip 26025  df-ph 26142
This theorem is referenced by:  ipasslem2  26161  ipasslem3  26162  ipasslem4  26163
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